{
  "abstract": "K-means is a widely adopted clustering approach in signal processing and machine learning. In this paper, we study K-means clustering through a cardinality-constrained formulation on a compact embedded submanifold. We replace the cardinality constraint with a difference-of-convex (DC) penalty and establish a global error bound to prove that the penalized and constrained formulations share the same global minimizers whenever the penalty parameter exceeds a finite threshold. To solve the resulting nonsmooth Riemannian DC problem, we reformulate it as a minimax problem and propose RADA-DC, a Riemannian alternating descent ascent method combining dual regularization with DC linearization. Under standard assumptions and suitable parameter choices, RADA-DC finds an $ε$-Riemannian critical point within $O(ε^{-3})$ iterations. We conduct experiments on synthetic and real-world datasets to demonstrate that the proposed method outperforms the tested baselines, including K-means++, in solution quality at competitive computational cost when the number of clusters is large.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34310",
  "author_total": 5,
  "authors": [
    {
      "name": "Meng Xu"
    },
    {
      "name": "Bo Jiang"
    },
    {
      "name": "Hanfu Zhang"
    },
    {
      "name": "Ya-Feng Liu"
    },
    {
      "name": "Anthony Man-Cho So"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c9gqwdvx3j3j2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34310",
  "fulltext_url": null,
  "id": "c9gqwdvx3j3j2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34310",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Advanced Clustering Algorithms Research"
    ]
  ],
  "openalex": "W7214762156",
  "pdf_url": "https://arxiv.org/pdf/2609.34310",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c9gqwdvx3j3j2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34310",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34310"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Riemannian Difference-of-Convex Optimization for K-Means Clustering",
  "title_text": "Riemannian Difference-of-Convex Optimization for K-Means Clustering",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/c9gqwdvx3j3j2",
  "venue": "arXiv",
  "versions": []
}